Car: 1991 Honda Accord
When I slowly accelerate from a stop through first and second there is a noticable vibration. There is no associated audible sound, just the feeling. About a third of the time the vibration occurs just before the shift from first to second, about a third of the time just after the shift, and about a third of the time its doesn't happen at all. If the vibration starts prior to the shift to second, the vibration stops after the shift to second. If the vibration starts after the shift to second, the vibration stops just prior to the shift to third.
A few days ago I heard a thump from the transmission as I accelerated from a stop. I started noticing the vibration about a day later.
I'm going to take to car to my local shop in the morning, but I'd like to hear any other ideas.

Well,....I spent 8 years as a transmission man (until I wised up and got out from underneath other peoples vehicles)
The problem with diagnosing a vibration without having the car there in person is that there are SO many things it could be, and everybody's description of the vibration is going to be different.
How many miles on the car?
Is the car an automatic, or standard?
If it is automatic, is it shifting too early or late? (it can "chatter" if it shifts to early, sort of feels like when you let out the clutch on a standard to early)
Are the motor mounts okay? (if the mounts are bad "particularly on a front drive" the torque of the shifts will cause the motor/tranny to rock back and forth)
Have you recently driven anywhere where your car may have contracted a case of gremlins?
These are just a few of the possibilities
Its really hard to say without driving and examining the car, take it to a reputable wrench, and hopefully he can help you.

It is an automatic with about 225,000 miles on it.
I checked the fluid. It's at the proper level and nice and clear. It's doesn't seem to shift any differently than before. In fact, there are no other symptoms other than the vibration. It's not very pronounced either. If it weren't my car I would not have noticed it.
My local shops are pretty trustworthy. At least I like to think that. I'll post the official diagnosis.

And the answer is....
A dirty exhaust gas recycle valve!
Apparently the valve was gummed up to the point where it would not function properly when the engine was under load, causing a cylinder to misfire. So why didn't it happen at higher speeds? It WAS happening, but at the higher speeds the road noise/vibration was able to cover it up. The tech disassembled the valve, cleaned it up, and presto!
Cost for diagnosis and repair: $140